Towards a $p$-adic Langlands programme

TL;DR

This paper discusses foundational aspects and recent developments in the p-adic Langlands program, aiming to deepen understanding of p-adic representations and their connections to arithmetic geometry.

Contribution

It provides an overview of the current state and future directions of the p-adic Langlands program, highlighting new conjectures and theoretical frameworks.

Findings

Summarizes key conjectures in p-adic Langlands theory

Proposes new approaches to p-adic representation classification

Connects p-adic Hodge theory with Langlands correspondences

Abstract

These are the notes for an eponymous course given by the authors at the summer school on p-adic arithmetic geometry in Hangzhou.